# Measurement of the Sensitivity Function in a Time-Domain Atomic Interferometer

**ABSTRACT** We present here an analysis of the sensitivity of a time-domain atomic interferometer to the phase noise of the lasers used to manipulate the atomic wave packets. The sensitivity function is calculated in the case of a three-pulse Mach-Zehnder interferometer, which is the configuration of the two inertial sensors we are building at the Laboratoire National de Metrologie et d'Essais-Systeme de References Temps-Espace. We successfully compare this calculation to experimental measurements. The sensitivity of the interferometer is limited by the phase noise of the lasers as well as by residual vibrations. We evaluate the performance that could be obtained with state-of-the-art quartz oscillators, as well as the impact of the residual phase noise of the phase-locked loop. Requirements on the level of vibrations are derived from the same formalism.

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**ABSTRACT:**The theory of general relativity describes macroscopic phenomena driven by the influence of gravity while quantum mechanics brilliantly accounts for microscopic effects. Despite their tremendous individual success, a complete unification of fundamental interactions is missing and remains one of the most challenging and important quests in modern theoretical physics. The STE-QUEST satellite mission, proposed as a medium-size mission within the Cosmic Vision program of the European Space Agency (ESA), aims for testing general relativity with high precision in two experiments by performing a measurement of the gravitational redshift of the Sun and the Moon by comparing terrestrial clocks, and by performing a test of the Universality of Free Fall of matter waves in the gravitational field of Earth comparing the trajectory of two Bose-Einstein condensates of Rb85 and Rb87. The two ultracold atom clouds are monitored very precisely thanks to techniques of atom interferometry. This allows to reach down to an uncertainty in the E\"otv\"os parameter of at least 2x10E-15. In this paper, we report about the results of the phase A mission study of the atom interferometer instrument covering the description of the main payload elements, the atomic source concept, and the systematic error sources.Classical and Quantum Gravity 01/2014; 31(11):115010. · 3.56 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We present a theoretical analysis of aliasing noises that might appear in cold atom Mach-Zehnder interferometers used for the measurement of various physical quantities. We focus more specifically on single cold atom gyroscopes. To evaluate the level of aliasing noises, we have developed a model based on the power spectral densities of the different identified noise sources as input parameters and which makes use of a servo-loop to realize a precise measurement of the rotation rate. The model allows one to take into account different modes of operation, like a continuous as well as a pulsed or even a multi-ball operation. For monokinetic atoms, we show that the intermodulation noise can be completely filtered out with a continuous mode of operation and an optimum modulation scheme for any modulation frequency but also with a pulsed operation however only for specific launching frequencies. In the case of a real continuous atomic beam having a velocity distribution, it comes out that a high attenuation can be reached which indicates clearly the potential stability improvement that can be expected from a continuous operation.The European Physical Journal D 66(3). · 1.51 Impact Factor - SourceAvailable from: Andrea BertoldiG. M. Tino, F. Sorrentino, D. Aguilera, B. Battelier, A. Bertoldi, Q. Bodart, K. Bongs, P. Bouyer, C. Braxmaier, L. Cacciapuoti, [......], A. Kubelka, A. Landragin, A. Milke, A. Peters, E. M. Rasel, E. Rocco, C. Schubert, T. Schuldt, K. Sengstock, A. Wicht[Show abstract] [Hide abstract]

**ABSTRACT:**Atom interferometry provides extremely sensitive and accurate tools for the measurement of inertial forces. Operation of atom interferometers in microgravity is expected to enhance the performance of such sensors. This paper presents two possible implementations of a dual 85 Rb- 87 Rb atom interferometer to perform differential gravity measurements in space, with the primary goal to test the Weak Equivalence Principle. The proposed scheme is in the framework of two projects of the European Space Agency, namely Q-WEP and STE-QUEST. The paper describes the baseline experimental conﬁguration, and discusses the technology readiness, noise and error budget for the two proposed experiments.Nuclear Physics B - Proceedings Supplements 01/2013; 243-244:203. · 0.88 Impact Factor

Page 1

arXiv:physics/0510197v1 [physics.atom-ph] 21 Oct 2005

SUBMITTED FOR PUBLICATION TO: IEEE TRANS. ON INSTRUM. MEAS., MARCH 24, 20051

Measurement of the sensitivity function in

time-domain atomic interferometer

P. Cheinet, B. Canuel, F. Pereira Dos Santos, A. Gauguet, F. Leduc, A. Landragin

Abstract

We present here an analysis of the sensitivity of a time-domain atomic interferometer to

the phase noise of the lasers used to manipulate the atomic wave-packets. The sensitivity

function is calculated in the case of a three pulse Mach-Zehnder interferometer, which is the

configuration of the two inertial sensors we are building at BNM-SYRTE. We successfully

compare this calculation to experimental measurements. The sensitivity of the interferometer

is limited by the phase noise of the lasers, as well as by residual vibrations. We evaluate

the performance that could be obtained with state of the art quartz oscillators, as well as

the impact of the residual phase noise of the phase-lock loop. Requirements on the level of

vibrations is derived from the same formalism.

Index Terms

Atom interferometry, Cold atoms, Sensitivity function, Stimulated Raman transition

I. Introduction

A

sensitivity inertial sensors [2], [3], [4], based on atomic interferometry [5], already reveal

accuracies comparable with state of the art sensors [6], [7]. Two cold atom inertial sensors

are currently under construction at BNM-SYRTE , a gyroscope [8] which already reaches a

sensitivity of 2.5 × 10−6rad.s−1.Hz−1/2, and an absolute gravimeter [9] which will be used

in the BNM Watt Balance project [10]. Although based on different atoms and geometries,

the atomic gyroscope and gravimeter rely on the same principle, which is presented in figure

1. Atoms are collected in a three dimensional magneto-optical trap (3D-MOT) in which the

atoms are cooled down to a few µK. In the gyroscope,133Cs atoms are launched upwards

with an angle of 8˚ with respect to verticality using the technic of moving molasses, whereas

in the gravimeter,

prepared by a combination of microwave and optical pulses. The manipulation of the atoms

is realized by stimulated Raman transition pulses [11], using two counter-propagating lasers,

which drive coherent transitions between the two hyperfine levels of the alkali atom. Three

laser pulses, of durations τR− 2τR− τR, separated in time by T, respectively split, redirect

and recombine the atomic wave-packets, creating an atomic interferometer [12]. Finally, a

fluorescence detection gives a measurement of the transition probability from one hyperfine

level to the other, which is given by P =1

The phase difference between the two Raman lasers (which we will call the Raman phase

throughout this article, and denote φ) is printed at each pulse on the phase of the atomic

TOM optics is a mean to realize precision measurements in various fields.

microwave clocks are the most precise realization of a SI unit, the second [1], and high

Atomic

87Rb atoms are simply let to fall. Then the initial quantum state is

2(1 − cos(Φ)), Φ being the interferometric phase.

P. Cheinet, B. Canuel, F. Pereira Dos Santos, A. Gauguet, F. Leduc and A. Landragin are with Laboratoire

BNM-SYRTE,75014 Paris, France (e-mail: patrick.cheinet@obspm.fr)

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CHEINET et al.: MEASUREMENT OF THE SENSITIVITY FUNCTION IN ...2

wave function [13]. As φ depends on the position of the atoms, the interferometer is sensitive

to inertial forces, and can thus measure rotation rates and accelerations. A drawback of this

technic is that the measurement of the interferometric phase is affected by the phase noise

of the Raman lasers, as well as parasitic vibrations. The aim of this article is to investigate

both theoretically and experimentally how these noise sources limit the sensitivity of such

an atomic interferometer.

Fig. 1

Scheme of principle of our inertial sensors, illustrated for the gyroscope experiment.

Cold atoms from the 3D-MOT are launched upwards and a pure quantum state is

selected. At the top of their trajectory, we apply three Raman laser pulses realizing

the interferometer. Finally a fluorescence detection allows to measure the transition

probability. Such an interferometer is sensitive to the rotation (Ω) perpendicular to the

area enclosed between the two arms and to the acceleration along the laser’s axis.

II. sensitivity function

The sensitivity function is a natural tool to characterize the influence of the fluctuations

in the Raman phase φ on the transition probability [14], and thus on the interferometric

phase. Let’s assume a phase jump δφ occurs on the Raman phase φ at time t during the

interferometer sequence, inducing a change of δP(δφ,t) in the transition probability. The

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CHEINET et al.: MEASUREMENT OF THE SENSITIVITY FUNCTION IN ...3

sensitivity function is then defined by :

g(t) = 2 lim

δφ→0

δP(δφ,t)

δφ

.(1)

The sensitivity function can easily be calculated for infinitesimally short Raman pulses.

In this case, the interferometric phase Φ can be deduced from the Raman phases φ1,φ2,φ3dur-

ing the three laser interactions, taken at the position of the center of the atomic wavepacket:

Φ = φ1− 2φ2+ φ3 [15]. Usually, the interferometer is operated at Φ = π/2, for which

the transition probability is 1/2, to get the highest sensitivity to interferometric phase

fluctuations. If the phase step δφ occurs for instance between the first and the second

pulses, the interferometric phase changes by δΦ = −δφ, and the transition probability by

δP = −cos(π/2 + δΦ)/2 ∼ −δφ/2 in the limit of an infinitesimal phase step. Thus, in

between the first two pulses, the sensitivity function is -1. The same way, one finds for the

sensitivity function between the last two pulses : +1.

In the general case of finite duration Raman laser pulses, the sensitivity function depends

on the evolution of the atomic state during the pulses. In order to calculate g(t), we make

several assumptions. First, the laser waves are considered as pure plane waves. The atomic

motion is then quantized in the direction parallel to the laser beams. Second, we restrict

our calculation to the case of a constant Rabi frequency (square pulses). Third, we assume

the resonance condition is fulfilled. The Raman interaction then couples the two states

|a? = |g1,− →p ? and |b? = |g2,− →p + ?− →

keff? where |g1? and |g2? are the two hyperfine levels

of the ground state,− →p is the atomic momentum,− →

vectors of the two lasers.

We develop the atomic wave function on the basis set {|a?,|b?} so that |Ψ(t)? = Ca(t)|a?+

Cb(t)|b?, and choose the initial state to be |Ψ(ti)? = |Ψi? = |a?. At the output of the

interferometer, the transition probability is given by P = |Cb(tf)|2, where tf= ti+2T +4τR.

The evolution of Caand Cbfrom tito tfis given by

keff is the difference between the wave

?

Ca(tf)

Cb(tf)

?

= M

?

Ca(ti)

Cb(ti)

?

(2)

where M is the evolution matrix through the whole interferometer. Solving the Schr¨ odinger

equation gives the evolution matrix during a Raman pulse[16], from time t0to time t:

Mp(t0,t,ΩR,φ) =

?

e−iωa(t−t0)cos(ΩR

−ie−iωb(t−t0)e−i(ωLt0+φ)sin(ΩR

2(t − t0))−ie−iωa(t−t0)ei(ωLt0+φ)sin(ΩR

e−iωb(t−t0)cos(ΩR

2(t − t0))

2(t − t0))

2(t − t0))

?

(3)

where ΩR/2π is the Rabi frequency and ωL, the effective frequency, is the frequency difference

between the two lasers, ωL = ω2− ω1. Setting ΩR = 0 in Mp(t0,t,ΩR,φ) gives the free

evolution matrix, which determines the evolution between the pulses. The evolution matrix

for the full evolution is obtained by taking the product of several matrices. When t occurs

during the i − th laser pulse, we split the evolution matrix of this pulse at time t into two

successive matrices, the first one with φi, and the second one with φ = φi+ δφ.

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CHEINET et al.: MEASUREMENT OF THE SENSITIVITY FUNCTION IN ...4

Finally, we choose the time origin at the middle of the second Raman pulse. We thus

have ti= −(T + 2τR) and tf = T + 2τR. We then calculate the change in the transition

probability for a infinitesimally small phase jump at any time t during the interferometer,

and deduce g(t). It is an odd function, whose expression is given here for t > 0:

g(t) =

?sin(ΩRt)

−sin(ΩR(T − t)) T + τR< t < T + 2τR

0 < t < τR

τR< t < T + τR

1

(4)

When the phase jump occurs outside the interferometer, the change in the transition

probability is null, so that g(t) = 0 for |t| > T + 2τR.

In order to validate this calculation, we use the gyroscope experiment to measure experi-

mentally the sensitivity function. About 108atoms from a background vapor are loaded in a

3D-MOT within 125 ms, with 6 laser beams tuned to the red of the F = 4 → F′= 5 transi-

tion at 852 nm. The atoms are then launched upwards at ∼ 2.4 m/s within 1 ms, and cooled

down to an effective temperature of ∼ 2.4µK. After launch, the atoms are prepared into the

|F = 3,mF= 0? state using a combination of microwave and laser pulses : they first enter a

selection cavity tuned to the |F = 4,mF= 0? → |F = 3,mF= 0? transition. The atoms left

in the F = 4 state are pushed away by a laser beam tuned to the F = 4 → F′= 5 transition,

11 cm above the selection cavity. The selected atoms then reach the apogee 245 ms after

the launch, where they experience three interferometer pulses of duration τR− 2τR− τR

with τR= 20 µs separated in time by T = 4.97 ms. The number of atoms NF=3and NF=4

are finally measured by detecting the fluorescence induced by a pair of laser beams located

7 cm below the apogee. From these measurements, we deduce the transition probability

NF=4/(NF=3+NF=4). The total number of detected atoms is about 105. The repetition rate

of the experiment is 2 Hz.

The set-up for the generation of the two Raman laser beams is displayed in figure 2.

Two slave diode lasers of 150 mW output power are injected with extended cavity diode

lasers. The polarizations of the slave diodes output beams are made orthogonal so that the

two beams can be combined onto a polarization beam splitter cube. The light at this cube

is then split in two distinct unbalanced paths.

On the first path, most of the power of each beam is sent through an optical fiber

to the vacuum chamber. The two beams are then collimated with an objective attached

onto the chamber (waist w0= 15 mm). They enter together through a viewport, cross the

atomic cloud, and are finally retroreflected by a mirror fixed outside the vacuum chamber.

In this geometry, four laser beams are actually sent onto the atoms, which interact with

only two of them, because of selection rules and resonance conditions. The interferometer

can also be operated with co-propagating Raman laser beams by simply blocking the light

in front of the retroreflecting mirror. A remarkable feature of this experiment is that the

three interferometer pulses are realized by this single pair of Raman lasers that is turned on

and off three times, the middle pulse being at the top of the atoms’ trajectory. For all the

measurements described in this article, the Raman lasers are used in the co − propagating

configuration. The interferometer is then no longer sensitive to inertial forces, but remains

sensitive to the relative phase of the Raman lasers. Moreover, as such Raman transitions

are not velocity selective, more atoms contribute to the signal. All this allows us to reach a

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CHEINET et al.: MEASUREMENT OF THE SENSITIVITY FUNCTION IN ...5

200 MHz?

LO?

reference?

9,? 3? 92..GHz? 9,? 3? 92..GHz?

9,192..GHz?

200 MHz?

IF?

Frequency?

Chain?

Master?

Laser 1 ?

Master?

Laser 2 ?

x2?

Slave?

Laser 2 ?

Slave?

Laser 1 ?

x94?

100 MHz ?

DRO?

Fig. 2

Principle of the laser phase-lock: the beatnote at 9.192 GHz between the two Raman

lasers is observed on a fast response photodetector. After amplification, this beatnote

is mixed with the reference frequency at 9.392 GHz from the frequency chain, to obtain a

signal at 200 MHz. This signal is compared with the reference frequency at 200 MHz from

the same frequency chain to get an error signal. This error signal is then processed and

sent to the current of the laser and to the PZT that controls the laser cavity length.

good signal to noise ratio of 150 per shot.

The second path is used to control the Raman lasers phase difference, which needs to be

locked [17] onto the phase of a very stable microwave oscillator. The phase lock loop scheme

is also displayed in figure2. The frequency difference is measured by a fast photodetector,

which detects a beatnote at 9.192 GHz. This signal is then mixed with the signal of a

Dielectric Resonator Oscillator (DRO) tuned at 9.392 GHz. The DRO itself is phase locked

onto the 94th harmonics of a very stable 100 MHz quartz. The output of the mixer (IF)

is 200 MHz. A local oscillator (LO) at 200 MHz is generated by doubling the same 100

MHz quartz. IF and LO are compared using a digital phase and frequency detector, whose

output is used as the error signal of the phase-locked loop. The relative phase of the lasers is

stabilized by reacting on the current of one of the two diode lasers, as well as on the voltage

applied to the PZT that controls the length of the extended cavity diode laser [17].

To measure g(t), a small phase step of δφ = 0.107 rad is applied at time t on the

local oscillator. The phase lock loop copies this phase step onto the Raman phase within

a fraction of µs, which is much shorter than the Raman pulse duration of τR = 20 µs.

Finally we measured the transition probability as a function of t and deduced the sensitivity

function. We display in figure 3 the measurement of the sensitivity function compared with

the theoretical calculation. We also realized a precise measurement during each pulse and

clearly obtained the predicted sinusoidal rise of the sensitivity function.

For a better agreement of the experimental data with the theoretical calculation, the

data are normalized to take into account the interferometer’s contrast, which was measured

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CHEINET et al.: MEASUREMENT OF THE SENSITIVITY FUNCTION IN ...6

?

?

?

?

?

?

?

?

?

Fig. 3

The atomic sensitivity function g(t) as a function of time, for a three pulses

interferometer with a Rabi frequency ΩR=

in solid line and the experimental measurement with crosses. A zoom is made on the first

pulse.

π

2τR. The theoretical calculation is displayed

to be 78%. This reduction in the contrast with respect to 100% is due to the combined effect

of inhomogeneous Rabi frequencies between the atoms, and unbalanced Rabi frequencies

between the pulses. Indeed, the atomic cloud size of 8 mm is not negligible with respect to

the size of the single pair of Raman gaussian beams, w0= 15mm. Atoms at both sides of

the atomic cloud will not see the same intensity, inducing variable transfer efficiency of the

Raman transitions. Moreover, the cloud moves by about 3 mm between the first and the

last pulse. In order for the cloud to explore only the central part of the gaussian beams,

we choose a rather small interaction time of T = 4.97 ms with respect to the maximum

interaction time possible of T = 40 ms. Still, the quantitative agreement is not perfect.

One especially observes a significant asymmetry of the sensitivity function, which remains

to be explained. A full numerical simulation could help in understanding the effect of the

experimental imperfections.

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CHEINET et al.: MEASUREMENT OF THE SENSITIVITY FUNCTION IN ...7

III. Transfer Function of the interferometer

From the sensitivity function, we can now evaluate the fluctuations of the interferometric

phase Φ for an arbitrary Raman phase noise φ(t) on the lasers

δΦ =

?+∞

−∞

g(t)dφ(t) =

?+∞

−∞

g(t)dφ(t)

dt

dt.(5)

The transfer function of the interferometer can be obtained by calculating the response

of the interferometer phase Φ to a sinusoidal modulation of the Raman phase, given by

φ(t) = A0cos(ω0t + ψ).We find δΦ = A0ω0Im(G(ω0))cos(ψ), where G is the Fourier

transform of the sensitivity function.

G(ω) =

?+∞

−∞

e−iωtg(t)dt(6)

When averaging over a random distribution of the modulation phase ψ, the rms value

of the interferometer phase is δΦrms= |A0ω0G(ω0)|. The transfer function is thus given

by H(ω) = ωG(ω). If we now assume uncorrelated Raman phase noise between successive

measurements, the rms standard deviation of the interferometric phase noise σrms

by:

?+∞

where Sφ(ω) is the power spectral density of the Raman phase.

We calculate the Fourier transform of the sensitivity function and find:

Φ

is given

(σrms

Φ )2=

0

|H(ω)|2Sφ(ω)dω(7)

G(ω) =

4iΩR

ω2− Ω2

R

sin(ω(T + 2τR)

2

)(cos(ω(T + 2τR)

2

) +ΩR

ω

sin(ωT

2))

(8)

At low frequency, where ω << ΩR, the sensitivity function can be approximated by

G(ω) = −4i

ωsin2(ωT/2) (9)

The weighting function |H(2πf)|2versus the frequency f is displayed in figure 4. It

has two important features: the first one is an oscillating behavior at a frequency given by

1/(T + 2τR), leading to zeros at frequencies given by fk=

first order filtering due to the finite duration of the Raman pulses, with an effective cutoff

frequency f0, given by f0=

3

displayed on the figure.

In order to measure the transfer function, a phase modulation Amcos(2πfmt + ψ) is

applied on the Raman phase, triggered on the first Raman pulse. The interferometric phase

variation is then recorded as a function of fm. We then repeat the measurements for the phase

modulation in quadrature Amsin(2πfmt+ψ). From the quadratic sum of these measurement,

we extract H(2πfm)2. The weighting function was first measured at low frequency. The

results, displayed in figure 5 together with the theoretical value, clearly demonstrate the

k

T+2τR. The second is a low pass

√3

ΩR

2π. Above 1 kHz only the mean value over one oscillation is

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CHEINET et al.: MEASUREMENT OF THE SENSITIVITY FUNCTION IN ...8

?

?

?

?

Fig. 4

Calculated weighting function for the Raman phase noise as a function of frequency.

Below 1 kHz, the exact weighting function is displayed. It shows an oscillation with a

period frequency of δf =

T+2τ. Above 1 kHz only the mean value of the weighting

function over δf is displayed. The weighting function acts as a first order low pass

filter, with an effective cutoff frequency of f0=

1

√3

3

ΩR

2π

oscillating behavior of the weighting function. Figure 6 displays the measurements performed

slightly above the cutoff frequency, and shows two zeros. The first one corresponds to a

frequency multiple of 1/(T + 2τ). The second one is a zero of the last factor of equation 8.

Its position depends critically on the value of the Rabi frequency.

When comparing the data with the calculation, the experimental imperfections already

mentioned have to be accounted for. An effective Rabi frequency Ωeffcan be defined by the

relation Ωeffτ0= π, where τ0is the duration of the single pulse, performed at the center

of the gaussian Raman beams, that optimizes the transition probability. For homogeneous

Raman beams, this pulse would be a π pulse. This effective Rabi frequency is measured with

an uncertainty of about 1 %. It had to be corrected by only 1.5 % in order for the theoretical

and experimental positions of the second zero to match. The excellent agreement between

the theoretical and experimental curves validate our model.

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CHEINET et al.: MEASUREMENT OF THE SENSITIVITY FUNCTION IN ...9

Fig. 5

The phase noise weighting function |H(2πf)2| for T = 4.97ms and τR= 20µs, at low

frequency. The theoretical calculation is displayed in solid line and the experimental

results in squares. We clearly see the oscillating behavior of the weighting function and

the experimental measurement are in good agreement with the theoretical calculation.

IV. Link between the sensitivity function and the sensitivity of the

interferometer

The sensitivity of the interferometer is characterized by the Allan variance of the inter-

ferometric phase fluctuations, σ2

Φ(τ), defined as

σ2

Φ(τ) =1

2?(¯ δΦk+1−¯ δΦk)2?

=1

2lim

n→∞

(10)

?

1

n

n

?

k=1

(¯ δΦk+1−¯ δΦk)2

?

. (11)

where ¯ δΦk is the average value of δΦ over the interval [tk,tk+1] of duration τ. The

Allan variance is equal, within a factor of two, to the variance of the differences in the

successive average values¯ δΦkof the interferometric phase. Our interferometer being operated

sequentially at a rate fc= 1/Tc, τ is a multiple of Tc: τ = mTc. Without loosing generality,

we can choose tk= −Tc/2 + kmTc. The average value¯ δΦkcan now be expressed as

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CHEINET et al.: MEASUREMENT OF THE SENSITIVITY FUNCTION IN ...10

Fig. 6

The phase noise weighting function |H(2πf)2| for T = 4.97ms and τR= 20µs,displayed near

the Rabi frequency. The theoretical calculation is displayed in solid line and the

experimental results in squares. We identified the zero multiple of

experimentally both zeros with a good agreement with theory.

1

T+2τand observed

¯ δΦk =

1

m

m

?

?tk+1

i=1

δΦi=1

m

m

?

i=1

?tk+iTc

tk+(i−1)Tc

g(t − tk− (i − 1)Tc− Tc/2)dφ

dtdt

(12)

=

1

m

tk

gk(t)dφ

dtdt

(13)

where gk(t) =?m

i=1g(t−kmTc−(i−1)Tc). The difference between successive average values

is then given by

¯ δΦk+1−¯ δΦk=1

m

?+∞

−∞

(gk+1(t) − gk(t))dφ

dtdt

(14)

For long enough averaging times, the fluctuations of the successive averages are not

correlated and the Allan variance is given by

σ2

Φ(τ) =1

2

1

m2

?+∞

0

|Gm(ω)|2ω2Sφ(ω)dω (15)

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CHEINET et al.: MEASUREMENT OF THE SENSITIVITY FUNCTION IN ...11

where Gmis the Fourier transform of the function gk+1(t) − gk(t). After a few algebra, we

find for the squared modulus of Gmthe following expression

|Gm(ω)|2= 4sin4(ωmTc/2)

sin2(ωTc/2)|G(ω)|2

(16)

When τ → ∞, |Gm(ω)|2∼2m

τ, the Allan variance of the interferometric phase is given by

Tc

?∞

j=−∞δ(ω − j2πfc)|G(ω)|2. Thus for large averaging times

σ2

Φ(τ) =1

τ

∞

?

n=1

|H(2πnfc)|2Sφ(2πnfc)(17)

Equation 17 shows that the sensitivity of the interferometer is limited by an aliasing phe-

nomenon similar to the Dick effect in atomic clocks[14] : only the phase noise at multiple of

the cycling frequency appear in the Allan variance, weighted by the Fourier components of

the transfer function.

Let’s examine now the case of white Raman phase noise : Sφ(ω) = S0

eter sensitivity is given by:

Φ(τ) = (π

φ. The interferom-

σ2

2)2S0

φ

τ

Tc

τR

(18)

In that case, the sensitivity of the interferometer depend not only on the Raman phase noise

spectral density but also on the pulse duration τR. For a better sensitivity, one should use

the largest pulse duration as possible. But, as the Raman transitions are velocity selective,

a very long pulse will reduce the number of useful atoms. This increases the detection noise

contribution, so that there is an optimum value of τR that depends on the experimental

parameters. In the case of the gyroscope, the optimum was found to be τR= 20µs.

To reach a good sensitivity, the Raman phase needs to be locked to the phase of a

very stable microwave oscillator (whose frequency is 6.834 GHz for87Rb and 9.192 GHz for

133Cs). This oscillator can be generated by a frequency chain, where low phase noise quartz

performances are transposed in the microwave domain. At low frequencies (f < 10−100 Hz),

the phase noise spectral density of such an oscillator is usually well approximated by a 1/f3

power law (flicker noise), whereas at high frequency (f > 1 kHz), it is independent of the

frequency (white noise). Using equation 17 and the typical parameters of our experiments

(τR= 20µs and T = 50 ms), we can calculate the phase noise spectral density required to

achieve an interferometric phase fluctuation of 1 mrad per shot. This is equivalent to the

quantum projection noise limit for 106detected atoms. The flicker noise of the microwave

oscillator should be lower than −53dB.rad2.Hz−1at 1 Hz from the carrier frequency, and

its white noise below −111dB.rad2.Hz−1. Unfortunately, there exists no quartz oscillator

combining these two levels of performance.

MHz oscillator (from Wenzel Company) onto a low flicker noise 5 MHz Blue Top oscillator

(Wenzel). From the specifications of these quartz, we calculate a contribution of 1.2 mrad

to the interferometric phase noise.

Phase fluctuations also arise from residual noise in the servo-lock loop. We have mea-

sured experimentally the residual phase noise power spectral density of a phase lock system

Thus, we plan to lock a SC Premium 100

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CHEINET et al.: MEASUREMENT OF THE SENSITIVITY FUNCTION IN ...12

analogous to the one described in figure 2. This system has been developed for phase locking

the Raman lasers of the gravimeter experiment. The measurement was performed by mixing

IF and LO onto an independent RF mixer, whose output phase fluctuations was analyzed

onto a Fast Fourier Transform analyzer. The result of the measurement is displayed on figure

7. At low frequencies, below 100 Hz, the phase noise of our phaselock system lies well below

the required flicker noise. After a few kHz, it reaches a plateau of −119dB.rad2.Hz−1. The

amplitude of this residual noise is not limited by the gain of the servo loop. Above 60 kHz,

it increases up to −90dB.rad2.Hz−1at 3.5 MHz, which is the bandwidth of our servo lock

loop. Using equation 17, we evaluated to 0.72 mrad its contribution to the interferometer’s

phase noise.

?

?

Fig. 7

Phase noise power spectral density between the two phase locked diode lasers. Up to

100 kHz, we display the residual noise of the phaselock loop, obtained by measuring the

phase noise of the demodulated beatnote on a Fast Fourier Transform analyzer. There,

the phase noise of the reference oscillator is rejected. Above 100 kHz, we display the

phase noise measured directly on the beatnote observed onto a spectrum analyzer. In

this case, the reference oscillator phase noise limits the Raman phase noise to

1.5 × 10−11rad2.Hz−1. In doted line is displayed an extrapolation of the phase noise due to

the phase-lock loop alone between 100 kHz and 300 kHz.

Other sources of noise are expected to contribute, which haven’t been investigated here

: noise of the fast photodetector, phase noise due to the propagation of the Raman beams

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CHEINET et al.: MEASUREMENT OF THE SENSITIVITY FUNCTION IN ...13

in free space and in optical fibers [18].

V. The case of parasitic vibrations

The same formalism can be used to evaluate the degradation of the sensitivity caused by

parasitic vibrations in the usual case of counter-propagating Raman beams. As the two laser

beams are first overlapped before being sent onto the atoms, their phase difference is mostly

affected by the movements of a single optical element, the mirror that finally retro-reflects

them.

A displacement of this mirror by δz induces a Raman phase shift of keffδz. The sensi-

tivity of the interferometer is then given by

σ2

Φ(τ) =k2

eff

τ

∞

?

n=1

|H(2πnfc)|2Sz(2πnfc)(19)

where Sz(ω) is the power spectral density of position noise. Introducing the power spectral

density of acceleration noise Sz(ω), the previous equation can be written

σ2

Φ(τ) =k2

eff

τ

∞

?

n=1

|H(2πnfc)|2

(2πnfc)4

Sa(2πnfc)(20)

It is important to note here that the acceleration noise is severely filtered by the transfer

function for acceleration which decreases as 1/f4.

In the case of white acceleration noise Sa, and to first order in τR/T, the limit on the

sensitivity of the interferometer is given by :

σ2

Φ(τ) =k2

effT4

2

?2Tc

3T− 1

?Sa

τ

(21)

To put this into numbers, we now calculate the requirements on the acceleration noise of

the retroreflecting mirror in order to reach a sensitivity of 1 mrad per shot. For the typical

parameters of our gravimeter, the amplitude noise should lie below 10−8m.s−2.Hz−1/2. The

typical amplitude of the vibration noise measured on the lab floor is 2 × 10−7m.s−2.Hz−1/2

at 1 Hz and rises up to about 5 × 10−5m.s−2.Hz−1/2at 10 Hz. This vibration noise can

be lowered to a few 10−7m.s−2.Hz−1/2in the 1 to 100 Hz frequency band with a passive

isolation platform. To fill the gap and cancel the effect of vibrations, one could use the

method proposed in [18], which consists in measuring the vibrations of the mirror with a

very low noise seismometer and compensate the fluctuations of the position of the mirror by

reacting on the Raman lasers phase difference.

VI. Conclusion

We have here calculated and experimentally measured the sensitivity function of a three

pulses atomic interferometer. This enables us to determine the influence of the Raman phase

noise, as well as of parasitic vibrations, on the noise on the interferometer phase. Reaching

a 1 mrad shot to shot fluctuation requires a very low phase noise frequency reference, an

optimized phase lock loop of the Raman lasers, together with a very low level of parasitic

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CHEINET et al.: MEASUREMENT OF THE SENSITIVITY FUNCTION IN ...14

vibrations. With our typical experimental parameters, this would result in a sensitivity of

4×10−8rad.s−1.Hz−1/2for the gyroscope and of 1.5×10−8m.s−2.Hz−1/2for the gravimeter.

Improvements are still possible. The frequency reference could be obtained from an

ultra stable microwave oscillator, such as a cryogenic sapphire oscillator [19], whose phase

noise lies well below the best quartz available. Besides, the requirements on the phase noise

would be easier to achieve using atoms with a lower hyperfine transition frequency, such as

Na or K. Trapping a very large initial number of atoms in the 3D-MOT would enable a

very drastic velocity selection. The duration of the Raman pulses could then be significantly

increased, which makes the interferometer less sensitive to high frequency Raman phase

noise. The manipulation of the atoms can also be implemented using Bragg pulses [20], [21].

The difference in the frequencies of the two beams being much smaller, the requirements

on the relative phase stability is easy to achieve. In that case, a different detection method

needs to be implemented as atoms in both exit ports of the interferometer are in the same

internal state. Using ultracold atoms with subrecoil temperature, atomic wavepackets at

the two exit ports can be spatially separated, which allows for a simple detection based on

absorption imaging. Such an interferometer would benefit from the long interaction times

available in space to reach a very high sensitivity.

We also want to emphasize that the sensitivity function can also be used to calculate the

phase shifts arising from all possible systematic effects, such as the light shifts, the magnetic

field gradients and the cold atom collisions.

Acknowledgment

The authors would like to thank Andr´ e Clairon for fruitful discussions and careful reading

of the manuscript. This work was supported in part by BNM, CNRS, DGA and CNES.

BNM-SYRTE is Unit´ e Associ´ ee au CNRS, UMR 8630.

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